1. One-to-one and Inverse Functions
Digital Lesson
One-to-one and Inverse Functions

2. Definition: Functions
A function y = f (x) with domain D is one-to-one on D if and only if for every x1 and x2 in D, f (x1) = f (x2) implies that x1 = x2.
A function is a mapping from its domain to its range so that each element, x, of the domain is mapped to one, and only one, element, f (x), of the range.
A function is one-to-one if each element f (x) of the range is mapped from one, and only one, element, x, of the domain.
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Definition: Functions

3. Horizontal Line Test
A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point. x y 2
Example: The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7).
(0, 7)
(4, 7)
y = 7
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Horizontal Line Test

4. Example: Horizontal Line Test
Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one.
a) y = x3
b) y = x3 + 3x2 – x – 1 x y -4 4 8 x y -4 4 8
one-to-one
not one-to-one
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Example: Horizontal Line Test

5. Inverse relation x = |y| + 1
Every function y = f (x) has an inverse relation x = f (y).
Domain
Range x y
3 2 1
Function y = |x| + 1
Domain
Range
Inverse relation x = |y| + 1 x y
3 2 1
The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}.
x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}.
The inverse relation is not a function. It pairs 2 to both -1 and +1.
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Inverse Relation

6. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation.
Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}.
The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the original function.
The range of the inverse relation is the domain of the original function.
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Ordered Pairs

7. Example: Ordered Pairs
The graphs of a relation and its inverse are reflections in the line y = x.
Example: Find the graph of the inverse relation geometrically from the graph of f (x) = x y 2 -2
The ordered pairs of f are given by the equation
y = x
The ordered pairs of the inverse are given by
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Example: Ordered Pairs

8. Example: Inverse Relation Algebraically
To find the inverse of a relation algebraically, interchange x and y and solve for y.
Example: Find the inverse relation algebraically for the function f (x) = 3x + 2.
y = 3x + 2 Original equation defining f
x = 3y + 2 Switch x and y.
3y + 2 = x Reverse sides of the equation.
y = Solve for y.
To calculate a value for the inverse of f, subtract 2, then divide by 3.
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Example: Inverse Relation Algebraically

9. The inverse function of y = f (x) is written y = f -1(x).
For a function y = f (x), the inverse relation of f is a function if and only if f is one-to-one.
For a function y = f (x), the inverse relation of f is a function if and only if the graph of f passes the horizontal line test.
If f is one-to-one, the inverse relation of f is a function called the inverse function of f.
The inverse function of y = f (x) is written y = f -1(x).
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Inverse Function

10. Example: Determine Inverse Function
Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. y
y = f -1(x)
y = x
The graph of f passes the horizontal line test.
y = f(x) x
The inverse relation is a function.
Reflect the graph of f in the line y = x to produce the graph of f -1.
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Example: Determine Inverse Function

11. Composition of Functions
The inverse function is an “inverse” with respect to the operation of composition of functions.
The inverse function “undoes” the function, that is, f -1( f (x)) = x.
The function is the inverse of its inverse function, that is, f ( f -1(x)) = x.
Example: The inverse of f (x) = x3 is f -1(x) = 3
f -1( f(x)) = = x and f ( f -1(x)) = ( )3 = x. 3
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Composition of Functions

12. Example: Composition of Functions
Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1.
g( f(x)) = = = = x
f(g(x)) = 2g(x) – 1 = 2( ) – 1 = (x + 1) – 1 = x
It follows that g = f -1.
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Example: Composition of Functions